Advanced Integration-Tabular Method

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The Advanced Integration-Tabular Method is a systematic approach to perform integration by parts multiple times. It is particularly effective when dealing with integrals where one function is easy to integrate and the other is a polynomial that eventually becomes zero after differentiations. If the original integral reappears during the process, it indicates that the integration can be stopped, and the result can be solved for. Caution is advised when applying this method, especially with cyclic integrals, as they may complicate the process. Overall, the tabular method is a valuable technique for efficiently solving complex integrals.
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"Advanced" Integration-Tabular Method

When cannot I not use this method?
If the integral is cyclic is there a way to get around it?
Any other information would be nice
Thanks
 
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Tabular integration is just a systematic way to integrate by parts multiple times. If the original integral reappears (without all other terms having canceled) you can stop and solve for it. A useful case is if p(x)f(x) where f is easy to integrate and p(x) is a polynomial so it will become zero after some number of differentiations.

Usual examples include

\int e^{-s t}\cos(a t) \text{ dt}
\int (x^2+3x+1)\sin(t) \text{ dt}
\int t^7 e^{-t} \text{ dt}
 


Note that you should be careful when the original integral reappears. Try integrating 1/x using integration by parts, with u = 1/x and dv=dx
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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